IIRC, lead acid batteries are around 95-97% recyclable.
Li-ion batteries arent very complex... The cans are either steel or a mylar bag, the electrodes are copper and aluminum foils. The electrodes are graphite on one side, and nickel, cobalt, aluminum, and/or manganese with carbon on the other. The electrolyte is a liquid organic. The separator is a polymer.
The steel cans are easily recycled. The current collector foils are also easily removed once cells are deeply discharged. The electrolyte may or may not be recoverable, and will readily flash at ambient temperatures. The challenge will be to get the coated and calendared electrode materials, which will also contain the li-ions, Ni, Co, Mn, etc., and get it off of the current collector foils so they can be cleanly recycled, and then the powders processed to reclaim the important materials in there...
So a good portion of the Li-ion battery is very straightforward, and does not have any abnormal recycling that differs from any other industrial process.
From a recycling standpoint the important part is the cathode, since it usually contains nickel and / or cobalt which are toxic metals that should not be put in a landfill, and those elements are also relatively valuable.
But the exact cathode material varies and there is no standard for labelling the batteries other than "Lithium Ion". So for each battery received the recycler needs to determine the type of cathode material, or have a process that works for any type.